It is worth comparing the solutions obtained here to the
electrically charged rotating dilaton black string solutions
presented in [1]}. In the present work I have studied the
magnetic spinning dilaton string which produces a longitudinal
magnetic field in AdS spaces which is the correct one generalizing
of the magnetic string solution of Dias and Lemos in dilaton
theory [2]}, while in [1]} I constructed charged
rotating dilaton black string in AdS spaces which is the
generalization of the charged rotating string solutions of
[4]} in dilaton gravity. Although solution (REF ) of
the present paper is similar to Eq. (16) of Ref. [1]}
(except the sign of \(c\) ) and both solutions represent dilaton
string, however, there are some different between the magnetic
string and the electrically charged
dilaton black string solutions. First, the choice of the metric gauge \([g_{tt}-\rho ^{2}\)  and \((g_{\rho \rho })^{-1}g_{\phi \phi }]\)  in the magnetic case which is quite different
from the Schwarzschild gauge \([(g_{\rho \rho })^{-1}g_{tt}\)  and \(g_{\phi \phi }\rho ^{2}] \)  proposed in
[1]}. Second, the electrically charged dilaton black
strings have an essential singularity located at \(r=0\)  and also
have horizons, while the magnetic strings version presented here
have no curvature singularity and no horizon, but have a conic
geometry. Third, when the rotation parameter is nonzero, the
magnetic string has a net electric charge which is proportional to
the rotation parameter, while charged dilaton black string has
always an electric charge regardless of the rotation parameter.
